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Properties

Label 2-975-195.38-c1-0-46
Degree $2$
Conductor $975$
Sign $0.972 - 0.234i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.597 − 0.597i)2-s + (1.67 − 0.458i)3-s + 1.28i·4-s + (0.724 − 1.27i)6-s + (2.39 + 2.39i)7-s + (1.96 + 1.96i)8-s + (2.58 − 1.53i)9-s − 1.92·11-s + (0.589 + 2.14i)12-s + (−3.00 − 1.98i)13-s + 2.86·14-s − 0.225·16-s + (4.01 + 4.01i)17-s + (0.627 − 2.45i)18-s − 3.56·19-s + ⋯
L(s)  = 1  + (0.422 − 0.422i)2-s + (0.964 − 0.264i)3-s + 0.642i·4-s + (0.295 − 0.519i)6-s + (0.906 + 0.906i)7-s + (0.694 + 0.694i)8-s + (0.860 − 0.510i)9-s − 0.581·11-s + (0.170 + 0.620i)12-s + (−0.834 − 0.551i)13-s + 0.765·14-s − 0.0562·16-s + (0.973 + 0.973i)17-s + (0.147 − 0.578i)18-s − 0.817·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.972 - 0.234i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (818, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.972 - 0.234i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.99270 + 0.356122i\)
\(L(\frac12)\) \(\approx\) \(2.99270 + 0.356122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 + 0.458i)T \)
5 \( 1 \)
13 \( 1 + (3.00 + 1.98i)T \)
good2 \( 1 + (-0.597 + 0.597i)T - 2iT^{2} \)
7 \( 1 + (-2.39 - 2.39i)T + 7iT^{2} \)
11 \( 1 + 1.92T + 11T^{2} \)
17 \( 1 + (-4.01 - 4.01i)T + 17iT^{2} \)
19 \( 1 + 3.56T + 19T^{2} \)
23 \( 1 + (-3.88 + 3.88i)T - 23iT^{2} \)
29 \( 1 - 2.60T + 29T^{2} \)
31 \( 1 - 8.42iT - 31T^{2} \)
37 \( 1 + (5.96 + 5.96i)T + 37iT^{2} \)
41 \( 1 + 1.16T + 41T^{2} \)
43 \( 1 + (0.498 + 0.498i)T + 43iT^{2} \)
47 \( 1 + (-2.66 + 2.66i)T - 47iT^{2} \)
53 \( 1 + (-4.62 + 4.62i)T - 53iT^{2} \)
59 \( 1 + 3.80iT - 59T^{2} \)
61 \( 1 + 7.40T + 61T^{2} \)
67 \( 1 + (-7.88 - 7.88i)T + 67iT^{2} \)
71 \( 1 - 2.61T + 71T^{2} \)
73 \( 1 + (-8.89 + 8.89i)T - 73iT^{2} \)
79 \( 1 + 7.19iT - 79T^{2} \)
83 \( 1 + (11.9 + 11.9i)T + 83iT^{2} \)
89 \( 1 - 4.47iT - 89T^{2} \)
97 \( 1 + (5.19 + 5.19i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.20779090505356817024944616367, −8.802312908175207176671226933322, −8.433019187500822413359278277960, −7.76819860725134516487398077742, −6.89529725833466320286467899120, −5.40034353908870283905892609243, −4.65505330887920879185404383065, −3.48185569388774267966891170877, −2.62723449015425296223123472462, −1.82820725537666854815578350834, 1.26178338413845960650176808049, 2.52704761256876043076377380389, 3.93732373929460336732651085670, 4.73534091983892150431398333345, 5.32768155014133889024532733615, 6.81028437018097276100916227903, 7.45574544602501178660516803407, 8.069693038679698357110997818678, 9.291866275234561500412444460228, 9.919558088765467646779138944071

Graph of the $Z$-function along the critical line